# Are elementary matrices the matrix representations of corresponding elementary row operations?

I just get confused with the relationship between elementary row operations and corresponding elementary matrices, which the textbook just used the word "equivalent" to describe.

I understand left-multiplication to a $m \times n$ matrix, say A, by elementary matrices truly performs the corresponding row operations to A. Since elementary row operations satisfy the definition of linear transformation, I think they are linear transformations on $M_{m \times n}(F)$. My question is, are these elementary matrices the matrix representations for elementary row operations? My guess is no and the matrix representations should be a $mn\times mn$ matrix.

Is this correct? Thanks a lot.

• You should perhaps add an explanation/definition of what exactly you call "elementary matrix" to. In my definition yes: elementary matrices correspond to elementary operations on the rows (columns) of a matrix, or even to composition of more than one elem. operation, if we allow slightly more complex matrices to be considered elementary. – DonAntonio Feb 18 '16 at 18:49
• Yes, you are right. The elementary row operations correspond to elementary matrices in the way you describe, but the matrix representations for elementary row operations are $mn\times mn$ matrices. – user84413 Feb 18 '16 at 19:27
• @Joanpemo I think your definition is the same as mine. – Yuxi Han Feb 20 '16 at 15:48
• @user84413 Thanks! – Yuxi Han Feb 20 '16 at 15:48